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Theorem xpen 7699
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpen  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )

Proof of Theorem xpen
StepHypRef Expression
1 relen 7540 . . . . 5  |-  Rel  ~~
21brrelexi 5049 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 7561 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 7633 . . . 4  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
52, 3, 4syl2anr 478 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
61brrelex2i 5050 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 7561 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 7632 . . . 4  |-  ( ( B  e.  _V  /\  C  ~<_  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
96, 7, 8syl2an 477 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
10 domtr 7587 . . 3  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  C )  /\  ( B  X.  C )  ~<_  ( B  X.  D ) )  ->  ( A  X.  C )  ~<_  ( B  X.  D ) )
115, 9, 10syl2anc 661 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  D ) )
121brrelex2i 5050 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 7583 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 7561 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 16 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 7633 . . . 4  |-  ( ( D  e.  _V  /\  B  ~<_  A )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
1712, 15, 16syl2anr 478 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
181brrelexi 5049 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 7583 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 7561 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 16 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 7632 . . . 4  |-  ( ( A  e.  _V  /\  D  ~<_  C )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
2318, 21, 22syl2an 477 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
24 domtr 7587 . . 3  |-  ( ( ( B  X.  D
)  ~<_  ( A  X.  D )  /\  ( A  X.  D )  ~<_  ( A  X.  C ) )  ->  ( B  X.  D )  ~<_  ( A  X.  C ) )
2517, 23, 24syl2anc 661 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  C ) )
26 sbth 7656 . 2  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  D )  /\  ( B  X.  D )  ~<_  ( A  X.  C ) )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
2711, 25, 26syl2anc 661 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   _Vcvv 3109   class class class wbr 4456    X. cxp 5006    ~~ cen 7532    ~<_ cdom 7533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537
This theorem is referenced by:  map2xp  7706  unxpdom2  7747  sucxpdom  7748  xpnum  8349  infxpenlem  8408  infxpidm2  8411  xpcdaen  8580  mapcdaen  8581  pwcdaen  8582  cdaxpdom  8586  ackbij1lem5  8621  canthp1lem1  9047  xpnnen  13954  xpomenOLD  13956  qnnen  13959  rexpen  13973  met2ndci  21151  re2ndc  21432  dyadmbl  22135  opnmblALT  22138  mbfimaopnlem  22188  mblfinlem1  30256
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